Lattice Chern-Simons theory has been studied for decades[1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}. Gauge invariance and duplication are the two major problems that prevent us from having a satistory lattice implementation of Chern-Simons theory with Abelian gauge group. For proposals that don't intruduce the dual lattice, the straightforward idea of replacing differentiation with lattice difference fails, because forward difference turns into backward difference under summation by parts, and the theory is no longer gauge invariant. One either has to invent very complicated versions of lattice difference [5]}[6]}, or introduce extra terms in the action to make it gauge invariant [9]}. Introducing the dual lattice solves the forward/backward difference problem and the theory is gauge invariant, but extra gauge fields and matter fields have to be introduced. The theory reduces to \(adb\)  Chern-Simons theory rather than \(ada\)  theory in the small lattice spacing limit [2]}. In reference [8]}, a straightforward construction is provided using the algebraic geometry language. However, this method only applies to simplex lattice and it cannot be generalized. Morever, the lattice action only agrees with the Abelian Chern-Simons action in continuous spacetime at leading order in the small lattice spacing limit. In reference [1]}, a very general construction of Abelian Chern-Simons theory on any proper spacial lattices is proposed. However, the time direction has to be continuous, and the action is not symmetric with respect to the three dimensions, which makes the calculation of Wilson loops that involve time components complicated. In this paper, a new definition of lattice Abelian Chern-Simons theory is proposed. This proposal introduces both lattice and dual lattice, and the gauge field on the dual lattice is expressed in terms of the gauge field on the original lattice. This treatment circumvents the issue of forward/backward difference, and also avoids the duplication problem, which prevents people from introducing the dual lattice. The mattter field can be regarded as a product of the parton fields on the vertexes of a lattice cell, and the Wilson loop consists of multiple lines on the lattice. This regularizes the divergence in the continuous theory. The form of the action is symmetric with respect to the three spacetime dimensions, and the calculation of Wilson loops is very straightforward.
